Abstract
In this paper, we consider the stochastic fractional-space Kuramoto–Sivashinsky equation forced by multiplicative noise. To obtain the exact solutions of the stochastic fractional-space Kuramoto–Sivashinsky equation, we apply the G′G-expansion method. Furthermore, we generalize some previous results that did not use this equation with multiplicative noise and fractional space. Additionally, we show the influence of the stochastic term on the exact solutions of the stochastic fractional-space Kuramoto–Sivashinsky equation.
Highlights
Fractional derivatives have received a lot of attention because they have been effectively used to problems in finance [1,2,3], biology [4], physics [5,6,7,8], thermodynamic [9,10], hydrology [11,12], biochemistry and chemistry [13]
Since fractionalorder integrals and derivatives allow for the representation of the memory and heredity properties of various substances, these new fractional-order models are more suited than the previously used integer-order models [14]
We presented different exact solutions of the stochastic fractional-space
Summary
The order α of Jumarie’s derivative is defined by [38]:. [ g(n) ( x )]α−n , n ≤ α ≤ n + 1, n ≥ 1, where g :R → R is a continuous function but not necessarily first-order differentiable and. The order α of Jumarie’s derivative is defined by [38]:. [ g(n) ( x )]α−n , n ≤ α ≤ n + 1, n ≥ 1, where g :R → R is a continuous function but not necessarily first-order differentiable and. Let us state some significant properties of modified Riemann–Liouville derivative as follows:. Dxα [ ag( x )] = aDxα g( x ), Dxα [ a f ( x ) + bg( x )] = aDxα f ( x ) + bDxα g( x ), and. D u, du x where σx is called the sigma indexes [39,40]
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